Given $ m \angle RPS = 2x + 25$, $ m \angle QPR = 6x - 16$, and $ m \angle QPS = 57$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {6x - 16} + {2x + 25} = {57}$ Combine like terms: $ 8x + 9 = 57$ Subtract $9$ from both sides: $ 8x = 48$ Divide both sides by $8$ to find $x$ $ x = 6$ Substitute $6$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 6({6}) - 16$ Simplify: $ {m\angle QPR = 36 - 16}$ So ${m\angle QPR = 20}$.